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Chapter 4: Problem 76

Find the domain of each logarithmic function. $$f(x)=\log _{5}(x+6)$$

Short Answer

Expert verified
The domain of the function is \(x > -6\).

Step by step solution

01

Set up the inequality

Considering the properties of logarithms, to find the domain of the function we need to set up an inequality where \(x + 6 > 0\). This is because a logarithm is defined only for positive numbers.
02

Solve the inequality

Solving the inequality \(x + 6 > 0\) for x, we would subtract 6 from both sides, which gives us \(x > -6\).
03

The Domain

With \(x > -6\), we find that the domain of the given function is all real numbers greater than -6.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can solve \(4^{x}=15\) by writing the equation in logarithmic form.

Use the exponential growth model, \(A=A_{0} e^{k t},\) to show that the time it takes a population to double (to grow from \(A_{0}\) to \(\left.2 A_{0}\right)\) is given by \(t=\frac{\ln 2}{k}\)

Find the domain of each logarithmic function. $$f(x)=\log \left(\frac{x+1}{x-5}\right)$$

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution=set. Verify this value by direct substitution into the equation. $$\log _{3}(4 x-7)=2$$

Explain how to find the domain of a logarithmic function.
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